A realistic theory of $\mathrm{E_{6}}$ unification through novel intermediate symmetries

Abstract

We propose a non-supersymmetric $\mathrm{E}_{6}$ GUT with the scalar sector consisting of $\mathbf{650}\oplus \mathbf{351'} \oplus \mathbf{27}$. Making use of the first representation for the initial symmetry breaking to an intermediate stage, and the latter two representations for second-stage breaking to the Standard Model and a realistic Yukawa sector, this theory represents the minimal $\mathrm{E}_{6}$ GUT that proceeds through one of the intermediate stages that are novel compared to $\mathrm{SU(5)}$ or $\mathrm{SO}(10)$ GUT: trinification $\mathrm{SU}(3)_C\times \mathrm{SU}(3)_L\times \mathrm{SU}(3)_R$, $\mathrm{SU}(6)\times \mathrm{SU}(2)$ and flipped $\mathrm{SO}(10)\times\mathrm{U}(1)$. We analyze these possibilities under the choice of vacuum that preserves a $\mathbb{Z}_{2}$ ``spinorial parity'', which disentangles the chiral and vector-like fermions of $\mathrm{E}_{6}$ and provides a dark matter candidate in the form of a (scalar) inert doublet. Three cases are shown to consistently unify under the extended survival hypothesis (with minimal fine-tuning): trinification symmetry $\mathrm{SU}(3)_C\times \mathrm{SU}(3)_L\times \mathrm{SU}(3)_R$ with either $LR$ or $CR$ parity, and $\mathrm{SU}(6)_{CR}\times\mathrm{SU}(2)_L$. Although the successful cases give a large range for proton lifetime estimates, all of them include regions consistent with current experimental bounds and within reach of forthcoming experiments. The scenario investigated in this paper essentially represents the unique (potentially) viable choice in the class of $\mathrm{E}_{6}$ GUTs proceeding through a novel-symmetry intermediate stage, since non-minimal alternatives seem to be intrinsically non-perturbative.